2. Multi-Valued Neurons and Multilayer Neural Network based on Multi-Valued Neurons MVN and MLMVN 1

3. A threshold function is a linearly separable function 1 1 f(1,1)= 1 f(1,-1)= -1 f(-1,1)= -1 f(-1,-1)= -1 f (x 1, x 2 ) is the OR function Linear separability means that it is possible to separate “1”s and “-1”s by a hyperplane 2

4. Threshold Boolean Functions The threshold (linearly separable) function can be learned by a single neuron The number of threshold functions is very small in comparison to the number of all functions (104 of 256 for n =3, about 2000 of 65536 for n =4, etc.) Non-threshold (nonlinearly separable) functions can not be learned by a single neuron (Minsky-Papert, 1969), they can be learned only by a neural network 3

5. XOR – a classical non-threshold (non-linearly separable) function 1 1 f(1,1)=1 f(1,-1)= -1 f(-1,1)= -1 f(-1,-1)=1 Non-linear separability means that it is impossible to separate “1”s and “-1”s by a hyperplane 4

6. Multi-valued mappings The first artificial neurons could learn only Boolean functions. However, the Boolean functions can describe only very limited class of problems. Thus, the ability to learn and implement not only Boolean, but also multiple-valued and continuous functions is very important for solving pattern recognition, classification and approximation problems. This determines the importance of those neurons that can learn and implement multiple-valued and continuous mappings 5

7. Traditional approach to learn the multiple-valued mappings by a neuron: Sigmoid activation function (the most popular): 1 0.5 6

8. Sigmoidal neurons: limitations Sigmoid activation function has a limited plasticity and a limited flexibility. Thus, to learn those functions whose behavior is quite different in comparison with the one of the sigmoid function, it is necessary to create a network, because a single sigmoidal neuron is not able to learn such functions. 7

9. Is it possible to overcome the Minsky’s-Papert’s limitation for the classical perceptron? Yes !!! 8

10. We can overcome the Minsky’s-Papert’s limitation using the complex-valued weights and the complex activation function 9

11. Is it possible to learn XOR and Parity n functions using a single neuron? Any classical monograph/text book on neural networks claims that to learn the XOR function a network from at least three neurons is needed. This is true for the real-valued neurons and real-valued neural networks. However, this is not true for the complex-valued neurons !!! A jump to the complex domain is a right way to overcome the Misky-Papert’s limitation and to learn multiple-valued and Boolean nonlinearly separable functions using a single neuron. 10

12. NEURAL NETWORKS Traditional Neurons Neuro-Fuzzy NetworksComplex-Valued NeuronsGeneralizations of Sigmoidal Neurons Multi-Valued and Universal Binary Neurons Multi-Valued and Universal Binary Neurons 11

13. Complex numbers  Unlike a real number, which is geometrically a point on a line, a complex number is a point on a plane.  Its coordinates are called a real (Re, horizontal) and an imaginary (Im, vertical) parts of the number  i is an imaginary unity  r is the modulo (absolute value) of the number Algebraic form of a complex number r 12

14. Complex numbers Trigonometric and exponential (Euler’s) forms of a complex number A unit circle φ is the argument (phase in terms of physics) of a complex number 13

15. Complex numbers Complex-conjugated numbers 14

16. XOR problem n=2, m=4 – four sectors W=(0, 1, i) – the weighting vector 1 i -i 15

17. Parity 3 problem n=3, m=6 : 6 sectors W=(0, ε, 1, 1) – the weighting vector 1 1 1 ε 1 16

18. Multi-Valued Neuron (MVN) A Multi-Valued Neuron is a neural element with n inputs and one output lying on the unit circle, and with the complex-valued weights. The theoretical background behind the MVN is the Multiple-Valued ( k -valued) Threshold Logic over the field of complex numbers 17

19. Multi-valued mappings and multiple- valued logic We traditionally use Boolean functions and Boolean (two-valued) logic, to present two-valued mappings: To present multi-valued mappings, we should use multiple-valued logic 18

20. Multiple-Valued Logic: classical view The values of multiple-valued ( k -valued) logic are traditionally encoded by the integers {0,1, …, k-1} On the one hand, this approach looks natural. On the other hand, it presents only the quantitative properties, while it can not present the qualitative properties. 19

21. Multiple-Valued Logic: classical view For example, we need to present different colors in terms of multiple-valued logic. Let Red=0, Orange=1, Yellow=2, Green=3, etc. What does it mean? Is it true that Red<Orange<Yellow<Green ??! 20

22. Multiple-Valued ( k -valued) logic over the field of complex numbers To represent and handle both the quantitative properties and the qualitative properties, it is possible to move to the field of complex numbers. In this case, the argument (phase) may be used to represent the quality and the amplitude may be used to represent the quantity 21

23. Multiple-Valued ( k -valued) logic over the field of complex numbers primitive k th root of unity regular values of k-valued logic i 0 1 k-1 k-2  22  k-1 1 one-to-one correspondence The k th roots of unity are values of k-valued logic over the field of complex numbers 22

24. Important advantage In multiple-valued logic over the field of complex numbers all values of this logic are algebraically (arithmetically) equitable: they are normalized and their absolute values are equal to 1 In the example with the colors, in terms of multiple-valued logic over the field of complex numbers they are coded by the different phases. Hence, their quality is presented by the phase. Since the phase determines the corresponding frequency, this representation meats the physical nature of the colors. 23

25. Discrete-Valued ( k -valued) Activation Function i 0 1 k-2 Z j-1 J j+1 k-1 Function P maps the complex plane into the set of the k th roots of unity 24

26. Discrete-Valued ( k -valued) Activation Function k=16 25

27. Multi-Valued Neuron (MVN) f is a function of k-valued logic (k-valued threshold function) 26

28. MVN: main properties The key properties of MVN: – Complex-valued weights – The activation function is a function of the argument of the weighted sum – Complex-valued inputs and output that are lying on the unit circle ( k th roots of unity) – Higher functionality than the one for the traditional neurons (e.g., sigmoidal) – Simplicity of learning 27

29. MVN Learning Learning is reduced to movement along the unit circle No derivative is needed, learning is based on the error- correction rule - error, which completely determines the weights adjustment - Desired output - Actual output i q s 28

30. Learning Algorithm for the Discrete MVN with the Error-Correction Learning Rule W – weighting vector; X - input vector is a complex conjugated to X α r – learning rate (should be always equal to 1) r - current iteration; r+1 – the next iteration is a desired output (sector) is an actual output (sector) i q s 29

31. Continuous-Valued Activation Function Continuous-valued case ( k   ): Function P maps the complex plane into the unit circle Z i 1 30

32. Continuous-Valued Activation Function 31

33. Continuous-Valued Activation Function 32

34. Learning Algorithm for the Continuous MVN with the Error Correction Learning Rule W – weighting vector; X - input vector is a complex conjugated to X α r – a learning rate (should be always equal to 1) r - current iteration; r+1 – the next iteration Z – the weighted sum i is a desired output is an actual output - neuron’s error 33

35. Learning Algorithm for the Continuous MVN with the Error Correction Learning Rule W – weighting vector; X - input vector is a complex conjugated to X α r – a learning rate (should be always equal to 1) r - current iteration; r+1 – the next iteration Z – the weighted sum i is a desired output is an actual output - neuron’s error 34

36. A role of the factor 1/(n+1) in the Learning Rule - neuron’s error The weights after the correction: The weighted sum after the correction: - exactly what we are looking for 35

37. Self-Adaptation of the Learning Rate -is the absolute value of the weighted sum on the previous (r th ) iteration. is a self-adaptive part of the learning rate i |z| <1 |z| >1 1 1/|z r | is a self-adaptive part of the learning rate 36

38. Modified Learning Rules with the Self- Adaptive Learning Rate 1/|z r | is a self-adaptive part of the learning rate Discrete MVN Continuous MVN 37

39. Convergence of the learning algorithm It is proven that the MVN learning algorithm converges after not more than k! iterations for the k -valued activation function For the continuous MVN the learning algorithm converges with the precision λ after not more than (π/λ)! i terations because in this case it is reduced to learning in π/λ –valued logic. 38

40. MVN as a model of a biological neuron No impulses  Inhibition  Zero frequency Intermediate State  Medium frequency Excitation  High frequency  The State of a biological neuron is determined by the frequency of the generated impulses  The amplitude of impulses is always a constant 39

41. MVN as a model of a biological neuron 40

42. MVN as a model of a biological neuron Maximal inhibition Maximal excitation 2π2π 0 π Intermediate State 41

43. MVN as a model of a biological neuron Maximal inhibition Maximal excitation 2π2π 0 π 42

44. MVN: Learns faster Adapts better Learns even highly nonlinear functions Opens new very promising opportunities for the network design Is much closer to the biological neuron Allows to use the Fourier Phase Spectrum as a source of the features for solving different recognition/classification problems Allows to use hybrid (discrete/continuous) inputs/output 43